1. Field of the Invention
The invention relates to a sub-band separating/combining apparatus having a band-separating filter bank for converting a digital signal to a plurality of sub-band signals and a band-combining filter bank which receives the sub-band signals after processing or transmission thereof, for combining these to recover the original digital signal or a processed version of that signal
2. Description of the Related Art
There are various applications in which a digital signal is supplied to a set of filters of a band-separating filter bank (sometimes referred to as an analyzing filter bank) to be spectrally divided into a plurality of sub-band signals, i.e., respectively corresponding to different frequency bands, with the sample rate of each of the sub-band signals then being reduced by decimation (i.e., down-sampling). Processing or transmission of the resultant low-bandwidth sub-band signals can then be efficiently performed. After processing or transmission of the sub-band signals, they are supplied to a band-combining filter bank (sometimes referred to as a synthesizing or a reconstructing filter bank), to be each subjected to interpolation processing (i.e., up-sampling), then inputted to respective ones of a set of filters whose outputs are additively combined to recover the original digital signal (or a processed version of that signal).
A prior art example of such a combination of a band-separating filter bank and a band-combining filter bank is shown in FIG. 7. Here, a band-separating filter bank 701, a processing section 703 and a band-combining filter bank 702 successively operate on an input digital signal designated as x(n). The processing section 703 may for example perform such operations as data encoding/decoding, echo cancellation processing, etc.
The band-separating filter bank 701 divides the input digital signal x(n) into a total of M channels of sub-band signals, whose respective frequency bands will be numbered as bands 0 to (M−1) respectively. 710˜71n designate the respective band-separating filters of the filter bank 701, respectively corresponding to frequency bands 0 to (M−1), with their respective Z-transform transfer functions (referred to in the following simply as transfer functions) designated as G0(z)˜GM−1(z). The output sub-band signals from these filters 710˜71n are supplied to respective ones of a set of decimators 720˜72n, with the resultant down-sampled sub-band signals being supplied to the processing section 703. The band-combining filter bank 702 includes a set interpolators 740˜74n which respectively receive the processed sub-band signals produced from the processing section 703, while 730˜73n are band-combining filters respectively corresponding to the frequency bands 0˜M−1 and having respective transfer functions K0(z)˜KM−1(z), which receive the corresponding ones of the interpolated sub-band signals which are produced from the interpolators 740˜74n. 
The decimation and interpolation factor is indicated as D. That is to say, one in every D samples of a sub-band signal is selected by the decimation processing, while (D−1) fixed sample values (e.g., zero values) are inserted following each sample of a processed sub-band signal, by the interpolation processing.
It will be assumed that the number M of sub-bands into which the input digital signal is divided is identical to the aforementioned decimation and interpolation factor D.
The output sub-band signals from the filters 730˜73n are additively combined in an adder 704, to obtain a digital signal y(n) as the output signal.
This is a recovered version of the original digital signal (possibly modified as a result of the operation of the processing section 703). If it is assumed that the processing 703 section performs a type of processing such as echo cancellation, which requires the use of DFT filter banks for band separation and combining, then the respective transfer functions Gk(z) and Kk(z) of the k-th band-separating filter and k-th band-combining filter are expressed as follows by equations (1) and (2) respectively:Gk(z)=G0(zWMk)  (1) Kk(z)=WM−k K0(zWMk)  (2) 
Here, WMk=exp(−j2πk/M), with 0<k<M−1, and each of G0(z) and K0(z) represents the transfer function of the prototype filter of a DFT (Discrete Fourier Transform) filter bank. The term “prototype filter” as used herein in relation to a band-separating filter bank or band-combining filter bank signifies a low-pass filter which handles the lowest frequency band, such as filter 710 of the band-separating filter bank 701 in FIG. 7.
If however the processing section 703 performs processing which requires the use of cosine modulation filter banks as the band-separating filter bank and band-combining filter bank, then the respective transfer functions of the k-th band-separating filter and k-th band-combining filter are obtained as follows from equations (3) and (4) respectively:Gk(z)=ak*ckP(zW2M(k+1/2))+ak*ck*P(zW2M−(k+1/2))  (3) Kk(z)=ak*ckP(zW2M(k+1/2))+ak*ck*P(zW2M−(k+1/2))  (4) 
Where W2M=exp(−jπ/M), with 0<k<M−1, ak=exp(jθk), Ck=W2M(k+1/2)(N−1)/2, θk=(2k+1)π/4, N is the number of taps of the prototype filter, the * symbol indicates the complex conjugate, and P(z) designates the transfer function of the prototype filter of a cosine modulation filter bank.
In the prior art, a FIR low-pass filter having a symmetric impulse response is used as the prototype filter in such a type of filter bank. An example of such a symmetric impulse response is shown in FIG. 8.
However with such a prior art type of apparatus which uses a filter in which each of the prototype filters of the sub-band separating filter bank and sub-band combining filter bank is a FIR (finite impulse response) low-pass filter having a symmetric impulse response, designating the number of taps of such a prototype filter as N, an amount of delay will be produced by the operation of a filter bank that is equal to the total of the group delays of (N−1) taps. In many applications, such an amount of delay becomes a serious disadvantage. For example, if the processing section 703 in FIG. 7 performs echo canceller processing, then it is essential to minimize the sub-band separating and sub-band combining filter delays, in order to achieve a suitably high speed of control response together with stability of operation.